Question

If $A=\{3,4,5\},B=\{1,2,3\}$ then

• A. $n(A\times B)=n(B\times A)$
• B. Total number of relations from $A$ to $B$ is $2^8$
• C. $n\left(\right(A\times B)\times (B\times A\left)\right)=81$
• D. $(A\times B)\cap (B\times A)=\left\{\right(3,3\left)\right\}$

Answer 1

Answer: (A), (C), (D)

The correct options are
A $n(A\times B)=n(B\times A)$
C $n\left(\right(A\times B)\times (B\times A\left)\right)=81$
D $(A\times B)\cap (B\times A)=\left\{\right(3,3\left)\right\}$

$A=\{3,4,5\},B=\{1,2,3\}$

$A\times B=\left\{\right(3,1),(3,2),(3,3),(4,1),(4,2),(4,3),(5,1),(5,2),(5,3\left)\right\}$

$B\times A=\left\{\right(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5\left)\right\}$

$\text{(a)}$
$n(A\times B)=n\left(A\right)\times n\left(B\right)=3\times 3=9$
$n(B\times A)=n\left(B\right)\times n\left(A\right)=3\times 3=9$
$n(A\times B)=n(B\times A)$

$\text{(b)}$
Total number of relations from $A$ to $B$
$=2^{n\left(A\right)\times n\left(B\right)}=2^{3\times 3}=2^9$

$\text{(c)}$
$n(A\times B)\times (B\times A))=n(A\times B)\times n(B\times A)$
$=9\times 9=81$

$\text{(d)}(A\times B)\cap (B\times A)=\left\{\right(3,3\left)\right\}$